Question

Define a relation $R$ on $A=\{1,2,3,4\}$ as ${}_x{R}_y$ if $x$ divides $y$. $R$ is

• A. reflexive and transitive
• B. reflexive and symmetric
• C. symmetric and transitive
• D. equivalence

Answer 1

Answer: (A)

Given set $A=\{1,2,3,4\}$ and relation, $xRy$ if $x$ divides $y$.
$\Rightarrow$ Relation
$=\{(1,1),(2,2),(3,3),(4,4),(1,2),(1,3)(1,4),(2,4)\}$
Reflexive We have, $xRy\iff y/x$ for $x,y\in A$ For any $x\in A$, we have $x/x\Rightarrow xRx$
Thus, $xRx$ for all $x\in A.$ So, $R$ is reflexive on $A$.
Symmetry $R$ ia not symmetry because, if $y/x$, then $x$ may not divide $y.$ For example $4/2$ but $2/4$
Transitive, Let $x,y,z\in A$, such that $xRy$ and $yRz.$
Then, $xRy$ and $yRz\Rightarrow \frac{y}{x}$ and $\frac{z}{y}\Rightarrow \frac{z}{x}.$
So, $R$ is a transitive relation on $A$.